s_examples/global_oce_optim/global_oce_estimation.tex

% $Header: /u/gcmpack/manual/part3/case_studies/global_oce_estimation/global_oce_estimation.tex,v 1.12 2008/02/11 21:39:06 dfer Exp $
% $Name:  $

\section[Global Ocean State Estimation Example]{Global Ocean State Estimation at $4^\circ$ Resolution}
\label{www:tutorials}
\label{sect:eg-global_state_estimate}
\begin{rawhtml}
<!-- CMIREDIR:eg-global_state_estimate: -->
\end{rawhtml}
\begin{center}
(in directory: {\it verification/tutorial\_global\_oce\_optim/})
\end{center}

\subsection{Overview}

This experiment illustrates the optimization capacity of the MITgcm: here,
a high level description.

In this tutorial, a very simple case is used to illustrate the optimization
capacity of the MITgcm. Using an ocean configuration with realistic geography
and bathymetry on a $4\times4^\circ$ spherical polar grid, we estimate a
time-independent surface heat flux adjustment $Q_\mathrm{netm}$ that attempts
to bring the model climatology into consistency with observations (Levitus
dataset, \cite{lev:94a}). The files for this experiment can be found in the
verification directory under tutorial\_global\_oce\_optim.

This adjustment $Q_\mathrm{netm}$ (a 2D field only function of longitude and
latitude) is the control variable of an optimization problem. It is inferred
by an iterative procedure using an `adjoint technique' and a least-squares
method (see, for example, \cite{stam-etal:02} and \cite{fer-eta:05}).

The ocean model is run forward in time and the quality of the solution is
determined by a cost function, $J_1$, a measure of the departure of the model
climatology from observations:
\begin{equation}\label{cost_temp}
J_1=\frac{1}{N}\sum_{i=1}^N \left[ \frac{\overline{T}_i-\overline{T}_i^{lev}}{\sigma_i^T}\right]^2
\end{equation}
where $\overline{T}_i$ and $\overline{T}_i^{lev}$ are, respectively, the model
and observed potential temperature at each
grid point $i$. The differences are weighted by an {\it a priori} uncertainty
$\sigma_i^T$ on observations (as provided by \cite{lev:94a}). The error
$\sigma_i^T$ is only a function of depth and varies from 0.5 at the surface to
0.05~K at the bottom of the ocean, mainly reflecting the decreasing
temperature variance with depth (Fig. \ref{Error}a). A value of $J_1$ of
order 1 means that the model is, on average, within observational
uncertainties.

The cost function also places constraints on the adjustment to insure it is
"reasonable", i.e. of order of the uncertainties on the observed surface heat
flux: 
\begin{equation}
J_2 = \frac{1}{N} \sum_{i=1}^N \left[\frac{Q_\mathrm{netm}}{\sigma^Q_i} \right]^2
\end{equation}
where $\sigma^Q_i$ are the {\it a priori} errors on the observed heat flux as
estimated by Stammer et al. (2002) from 30\% of local root-mean-square
variability of the NCEP forcing field (Fig \ref{Error}b).

The total cost function is defined as $J=\lambda_1 J_1+ \lambda_2 J_2$ where
$\lambda_1$ and $\lambda_2$ are weights controlling the relative contribution
of the two components. The adjoint model then yields the sensitivities
$\partial J/\partial Q_\mathrm{netm}$ of $J$ relative to the 2D fields
$Q_\mathrm{netm}$. Using a line-searching algorithm (\cite{gil-lem:89}),
$Q_\mathrm{netm}$ is adjusted then in the sense to
reduce $J$ --- the procedure is repeated until convergence.

%The configuration is identical 
%to the ``Global Ocean circulation'' tutorial where more details can be found.

Fig. \ref{Results} shows the results of such an optimization. The
model is started from rest and from January-mean temperature and salinity
initial conditions taken from the Levitus dataset. The experiment is run a year
and the averaged temperature over the whole run (i.e. annual mean) is used
in the cost function (\ref{cost_temp}) to evaluate the model. Only the
top 2 levels are used. The first guess $Q_\mathrm{netm}$ is chosen to be
zero. The weights $\lambda_1$ and $\lambda_2$ are set to 1 and 2, respectively.
The total cost function converges after 15 iterations, decreasing from 6.1 to
2.7 (the temperature contribution decreases from 6.1 to 1.8 while the heat
flux one increases from 0 to 0.42). The right panels of Fig. (\ref{Results})
illustrate the evolution of the temperature error at the surface from
iteration 0 to iteration 15. Unsurprisingly, the largest errors at iteration 0
(up to 6$^\circ$C, top left panels) are found in the Western boundary
currents. After optimization, the departure of the model temperature from
observations is reduced to 1$^\circ$C or less almost everywhere except in the
Pacific Equatorial Cold Tongue. Comparison of the initial temperature
error (top, right) and heat flux adjustment (bottom, left) shows that the
system basically increased the heat flux out of the ocean where temperatures
were too warm and vice-versa. Obviously, heat flux uncertainties are not the
sole responsible for temperature errors and the heat flux adjustment partly
compensates the poor representation of narrow currents (Western boundary
currents, Equatorial currents) at $4\times4^\circ$ resolution. This is
allowed by the large {\it a priori} error on the heat flux (Fig. \ref{Error}).
The Pacific Cold Tongue is a counter example: there, heat fluxes uncertainties
are fairly small (about 20~W.m$^2$), and a large temperature errors 
remains after optimization.

In the following, section 2 describes in details the implementation of the 
control variable $Q_\mathrm{netm}$, the cost function $J$ and the I/O required
for the communication between the model and the line-search. Instructions to
compile the MITgcm and its adjoint and the line-search algorithm are given in
section 3. The method used to run the experiment is described in section 4.

\begin{figure} [tpb]
\begin{center}
\includegraphics[width=\textwidth,height=.3\textheight]{part3/case_studies/global_oce_estimation/Error.eps}
\caption{{\it A priori} errors on potential temperature (left, in $^\circ$C) and
surface heat flux (right, in W~m$^{-2}$) used to compute the cost
terms $J_1$ and $J_2$, respectively.} 
\label{Error}
\end{center}
\end{figure}

\begin{figure} [tpb]
\begin{center}
\includegraphics[width=\textwidth,height=.3\textheight]{part3/case_studies/global_oce_estimation/Tutorial_fig.eps}
\caption{Initial annual mean surface heat flux (top right in W.m$^{-2}$) and 
adjustment obtained at iteration 15 (bottom right). Averaged difference
between model and observed potential temperatures at the surface (in $^\circ$C) 
before optimization (iteration 0, top right) and after optimization
(iteration 15, bottom right). Contour intervals for heat flux and temperature
are 25~W.m$^{-2}$ and 1$^\circ$C, respectively. A positive flux is out of the
ocean.}
\label{Results}
\end{center}
\end{figure}

\subsection{Implementation of the control variable and the cost function}

One of the goal of this tutorial is to illustrate how to implement a new
control variable. Most of this is fairly generic and is done in the ctrl
and cost packages found in the pkg/ directory. The modifications can be
tracked by the CPP option ALLOW\_HFLUXM\_CONTROL or the comment
cHFLUXM\_CONTROL. The more specific modifications required for the experiment
are found in verification/tutorial\_global\_oce\_optim/code\_ad. Here follows
a brief description of the implementation.

\subsubsection{The control variable}

The adjustment $Q_\mathrm{netm}$ is activated by setting
ALLOW\_HFLUXM\_CONTROL to "define" in ECCO\_OPTIONS.h.

It is first implemented as a ``normal'' forcing variable. It is defined in
FFIELDS.h, initialized to zero in ini\_forcing.F, and then used in
external\_forcing\_surf.F. $Q_\mathrm{netm}$ is made a control variable in
the ctrl package by modifying the following subroutines:

\begin{itemize}
\item ctrl\_init.F where $Q_\mathrm{netm}$ is defined as the control variable
number 24,

\item ctrl\_pack.F which writes, at the end of each iteration, the sensitivity
of the cost function $\partial J/\partial Q_\mathrm{netm}$ in to a file to be
used by the line-search algorithm,

\item ctrl\_unpack.F which reads, at the start of each iteration, the updated
adjustment as provided by the line-search algorithm,

\item ctrl\_map\_forcing.F in which the updated adjustment is added to the
first guess $Q_\mathrm{netm}$.
\end{itemize}

Note also some minor changes in ctrl.h, ctrl\_readparams.F, and ctrl\_dummy.h
(xx\_hfluxm\_file, fname\_hfluxm, xx\_hfluxm\_dummy).

\subsubsection{Cost functions}

The cost functions are implemented using the {\it cost} package.

\begin{itemize}

\item The temperature cost function $J_1$ which measures the drift of the mean
model temperature from the Levitus climatology is implemented in cost\_temp.F.
It is activated by ALLOW\_COST\_TEMP in ECCO\_OPTIONS.h. It requires the mean
temperature of the model which is obtained by accumulating the temperature in
cost\_tile.F (called at each time step).
The value of the cost function is stored in {\it objf\_temp} and its weight
$\lambda_1$ in {\it mult\_temp}.

\item The heat flux cost function, penalizing the departure of the surface
heat flux from observations is implemented in cost\_hflux.F, and activated by
the key ALLOW\_COST\_HFLUXM in ECCO\_OPTIONS.h. The value of the cost
function is stored in {\it objf\_hfluxm} and its weight $\lambda_2$ in
{\it mult\_hfluxm}.

\item The subroutine cost\_final.F calls the cost\_functions subroutines
and make the (weighted) sum of the various contributions.

\item The various weights used in the cost functions are read in
cost\_weights.F. The weight of the cost functions are read in
cost\_readparams.F from the input file data.cost.    

\end{itemize}


\subsection{Code Configuration}
\label{www:tutorials}
\label{SEC:eg_fourl_code_config}

The model configuration for this experiment resides under the directory
{\it verification/tutorial\_global\_oce\_optim/}.  The experiment files in
code\_ad/ and input\_ad/ contain the code customizations and parameter
settings. Most of them are identical to those used in the Global Ocean
( experiment {\it  tutorial\_global\_oce\_latlon}). Below, we describe some of
the customizations required for this experiment.

\subsubsection{Compilation-time customizations in {\it code\_ad/}}

In ECCO\_CPPOPTIONS.h:

\begin{itemize}
\item define ALLOW\_ECCO\_OPTIMIZATION 

\item define ALLOW\_COST, ALLOW\_COST\_TEMP, and ALLOW\_COST\_HFLUXM

\item define ALLOW\_HFLUXM\_CONTROL
\end{itemize}

\subsubsection{Running-time customizations in {\it input\_ad/}}

\begin{itemize}

\item {\it data}: note the smaller {\it cg2dTargetResidual} than in the
forward-only experiment,

\item {\it data.optim} specifies the iteration number,

\item {\it data.ctrl} is used, in particular, to specify the
name of the sensitivity and adjustment files associated to a control
variable,

\item {\it data.cost}: parameters of the cost functions, in particular
{\it lastinterval} specifies the length of time-averaging for the model
temperature to be used in the cost function (\ref{cost_temp}),

\item {\it data.pkg}: note that the Gradient Check package is turned on by
default (useGrdchk=.TRUE.),

\item {\it Err\_hflux.bin} and {\it Err\_levitus\_15layer.bin} are the
files containing the heat flux and potential temperature uncertainties,
respectively.

\end{itemize}

\subsection{Compiling} 

The optimization experiment requires two executables: 1) the 
MITgcm and its adjoint ({\it mitgcmuv\_ad}) and 2) the line-search
algorithm ({\it optim.x}).

\subsubsection{Compilation of MITgcm and its adjoint: {\it mitcgmuv\_ad}}

Before compiling, first note that, in the directory code\_ad/, two files
must be updated:
\begin{itemize}
\item code\_ad\_diff.list which lists new subroutines to be compiled
by the TAF software (cost\_temp.F and cost\_hflux.F here),

\item  the adjoint\_hfluxm files which provides a list of the control variables
and the name of cost function to the TAF software.

\end{itemize}

Then, in the directory build\_ad/, type:
\begin{verbatim}
% ../../../tools/genmake2 -mods=../code\_ad -adof=../code\_ad/adjoint\_hfluxm
% make depend
% make adall
\end{verbatim}
to generate the MITgcm executable {\it mitgcmuv\_ad}.

\subsubsection{Compilation of the line-search algorithm: {\it optim.x}}

This is done from the directories lsopt/ and optim/ (under MITgcm/). In lsopt/,
unzip the blash1 library adapted to your platform, and change the Makefile
accordingly. Compile with:
\begin{verbatim}
% make all
\end{verbatim}
(more details in lsopt\_doc.txt)

In optim/,  the path of the directory where {\it mitgcm\_ad} was compiled
must be specified in the Makefile in the variable INCLUDEDIRS. The file name
of the control variable (xx\_hfluxm\_file here) must be added to the name list
read by optim\_num.F. Then use
\begin{verbatim}
% make depend
\end{verbatim} 
and
\begin{verbatim}
% make
\end{verbatim}
to generate the line-search executable {\it optim.x}.

\subsection{Running the estimation}

Copy the {\it mitgcmuv\_ad} executable to input\_ad/ and {\it optim.x} to the
subdirectory input\_ad/OPTIM/. Move into input\_ad/. The first iteration is
somewhat particular and is best done "by hand" while the following iterations
can be run automatically (see below). Check that the iteration number is set
to 0 in data.optim and run the MITgcm:
\begin{verbatim}
% ./mitgcmuv_ad
\end{verbatim}

The output files adxx\_hfluxm.0000000000.* and xx\_hfluxm.0000000000.* contain
the sensitivity of the cost function to $Q_\mathrm{netm}$ and the adjustment
to $Q_\mathrm{netm}$ (zero at the first iteration), respectively. Two other
files called costhflux\_tut\_MITgcm.opt0000 and ctrlhflux\_tut\_MITgcm.opt0000
are also generated. They essentially contain the same information as the
adxx\_.hfluxm* and xx\_hfluxm* files, but in a compressed format. These two files
are the only ones involved in the communication between the adjoint model
{\it mitgcmuv\_ad} and the line-search algorithm {\it optim.x}. Only at the first
iteration, are they both generated by {\it mitgcmuv\_ad}. Subsenquently,
costhflux\_tut\_MITgcm.opt$n$ is an output of the adjoint model at
iteration $n$ and an input of the line-search. The latter returns an updated
adjustment in ctrlhflux\_tut\_MITgcm.opt$n+1$ to be used as an input of the
adjoint model at iteration n+1. 

At the first iteration, move costhflux\_tut\_MITgcm.opt0000 and
ctrlhflux\_tut\_MITgcm.opt0000 to OPTIM/, move into this directory and link data.optim 
and data.ctrl locally:
\begin{verbatim}
% cd OPTIM/
% ln -s ../data.optim .
% ln -s ../data.ctrl .
\end{verbatim}
The target cost function {\it fmin} needs to be specified too: as a rule of thumb,
it should be about 0.95-0.90 times the value of the cost function at
the first iteration. This value is only used at the first iteration and does
not need to be updated afterward. However, it implicitly specifies the
``pace'' at which the cost function is going down (if you are lucky and it does
indeed diminish!). More in the ECCO section maybe ?

Once this is done, run the line-search algorithm:
\begin{verbatim}
% ./optim.x
\end{verbatim}
which computes the updated adjustment for iteration 1, ctrlhflux\_tut\_MITgcm.opt0001.

The following iterations can be executed automatically using the shell
script {\it cycsh} found in input\_ad/. This script will take care of changing
the iteration numbers in the data.optim, launch the adjoint model, clean and
store the outputs, move the costhflux* and ctrlhflux* files, and run the
line-search algorithm. Edit {\it cycsh} to specify the prefix of the
directories used to store the outputs and the maximum number of iteration.

1	jmc	1.13	% $Header: /u/gcmpack/manual/part3/case_studies/global_oce_estimation/global_oce_estimation.tex,v 1.12 2008/02/11 21:39:06 dfer Exp $
2	dfer	1.1	% $Name: $
3
4	dfer	1.5	\section[Global Ocean State Estimation Example]{Global Ocean State Estimation at $4^\circ$ Resolution}
5	dfer	1.1	\label{www:tutorials}
6			\label{sect:eg-global_state_estimate}
7			\begin{rawhtml}
8			<!-- CMIREDIR:eg-global_state_estimate: -->
9			\end{rawhtml}
10	jmc	1.9	\begin{center}
11			(in directory: {\it verification/tutorial\_global\_oce\_optim/})
12			\end{center}
13	dfer	1.1
14	dfer	1.3	\subsection{Overview}
15
16	dfer	1.12	This experiment illustrates the optimization capacity of the MITgcm: here,
17			a high level description.
18	dfer	1.1
19	dfer	1.12	In this tutorial, a very simple case is used to illustrate the optimization
20			capacity of the MITgcm. Using an ocean configuration with realistic geography
21			and bathymetry on a $4\times4^\circ$ spherical polar grid, we estimate a
22			time-independent surface heat flux adjustment $Q_\mathrm{netm}$ that attempts
23			to bring the model climatology into consistency with observations (Levitus
24			dataset, \cite{lev:94a}). The files for this experiment can be found in the
25			verification directory under tutorial\_global\_oce\_optim.
26	dfer	1.1
27	dfer	1.10	This adjustment $Q_\mathrm{netm}$ (a 2D field only function of longitude and
28			latitude) is the control variable of an optimization problem. It is inferred
29			by an iterative procedure using an `adjoint technique' and a least-squares
30	dfer	1.12	method (see, for example, \cite{stam-etal:02} and \cite{fer-eta:05}).
31	dfer	1.1
32			The ocean model is run forward in time and the quality of the solution is
33			determined by a cost function, $J_1$, a measure of the departure of the model
34			climatology from observations:
35	dfer	1.12	\begin{equation}\label{cost_temp}
36	dfer	1.1	J_1=\frac{1}{N}\sum_{i=1}^N \left[ \frac{\overline{T}_i-\overline{T}_i^{lev}}{\sigma_i^T}\right]^2
37			\end{equation}
38	dfer	1.12	where $\overline{T}_i$ and $\overline{T}_i^{lev}$ are, respectively, the model
39			and observed potential temperature at each
40	dfer	1.10	grid point $i$. The differences are weighted by an {\it a priori} uncertainty
41	dfer	1.12	$\sigma_i^T$ on observations (as provided by \cite{lev:94a}). The error
42	dfer	1.10	$\sigma_i^T$ is only a function of depth and varies from 0.5 at the surface to
43			0.05~K at the bottom of the ocean, mainly reflecting the decreasing
44			temperature variance with depth (Fig. \ref{Error}a). A value of $J_1$ of
45			order 1 means that the model is, on average, within observational
46			uncertainties.
47	dfer	1.1
48	dfer	1.10	The cost function also places constraints on the adjustment to insure it is
49			"reasonable", i.e. of order of the uncertainties on the observed surface heat
50	dfer	1.1	flux:
51			\begin{equation}
52	dfer	1.10	J_2 = \frac{1}{N} \sum_{i=1}^N \left[\frac{Q_\mathrm{netm}}{\sigma^Q_i} \right]^2
53	dfer	1.1	\end{equation}
54	dfer	1.10	where $\sigma^Q_i$ are the {\it a priori} errors on the observed heat flux as
55			estimated by Stammer et al. (2002) from 30\% of local root-mean-square
56	dfer	1.12	variability of the NCEP forcing field (Fig \ref{Error}b).
57	dfer	1.1
58	dfer	1.10	The total cost function is defined as $J=\lambda_1 J_1+ \lambda_2 J_2$ where
59	dfer	1.1	$\lambda_1$ and $\lambda_2$ are weights controlling the relative contribution
60	dfer	1.10	of the two components. The adjoint model then yields the sensitivities
61	dfer	1.1	$\partial J/\partial Q_\mathrm{netm}$ of $J$ relative to the 2D fields
62	dfer	1.12	$Q_\mathrm{netm}$. Using a line-searching algorithm (\cite{gil-lem:89}),
63			$Q_\mathrm{netm}$ is adjusted then in the sense to
64	dfer	1.10	reduce $J$ --- the procedure is repeated until convergence.
65
66	dfer	1.12	%The configuration is identical
67			%to the ``Global Ocean circulation'' tutorial where more details can be found.
68
69			Fig. \ref{Results} shows the results of such an optimization. The
70			model is started from rest and from January-mean temperature and salinity
71			initial conditions taken from the Levitus dataset. The experiment is run a year
72			and the averaged temperature over the whole run (i.e. annual mean) is used
73			in the cost function (\ref{cost_temp}) to evaluate the model. Only the
74			top 2 levels are used. The first guess $Q_\mathrm{netm}$ is chosen to be
75			zero. The weights $\lambda_1$ and $\lambda_2$ are set to 1 and 2, respectively.
76			The total cost function converges after 15 iterations, decreasing from 6.1 to
77			2.7 (the temperature contribution decreases from 6.1 to 1.8 while the heat
78			flux one increases from 0 to 0.42). The right panels of Fig. (\ref{Results})
79			illustrate the evolution of the temperature error at the surface from
80			iteration 0 to iteration 15. Unsurprisingly, the largest errors at iteration 0
81			(up to 6$^\circ$C, top left panels) are found in the Western boundary
82			currents. After optimization, the departure of the model temperature from
83			observations is reduced to 1$^\circ$C or less almost everywhere except in the
84			Pacific Equatorial Cold Tongue. Comparison of the initial temperature
85			error (top, right) and heat flux adjustment (bottom, left) shows that the
86			system basically increased the heat flux out of the ocean where temperatures
87			were too warm and vice-versa. Obviously, heat flux uncertainties are not the
88			sole responsible for temperature errors and the heat flux adjustment partly
89			compensates the poor representation of narrow currents (Western boundary
90			currents, Equatorial currents) at $4\times4^\circ$ resolution. This is
91			allowed by the large {\it a priori} error on the heat flux (Fig. \ref{Error}).
92			The Pacific Cold Tongue is a counter example: there, heat fluxes uncertainties
93			are fairly small (about 20~W.m$^2$), and a large temperature errors
94			remains after optimization.
95
96			In the following, section 2 describes in details the implementation of the
97			control variable $Q_\mathrm{netm}$, the cost function $J$ and the I/O required
98			for the communication between the model and the line-search. Instructions to
99			compile the MITgcm and its adjoint and the line-search algorithm are given in
100			section 3. The method used to run the experiment is described in section 4.
101	dfer	1.10
102			\begin{figure} [tpb]
103			\begin{center}
104			\includegraphics[width=\textwidth,height=.3\textheight]{part3/case_studies/global_oce_estimation/Error.eps}
105	dfer	1.12	\caption{{\it A priori} errors on potential temperature (left, in $^\circ$C) and
106			surface heat flux (right, in W~m$^{-2}$) used to compute the cost
107			terms $J_1$ and $J_2$, respectively.}
108	dfer	1.10	\label{Error}
109			\end{center}
110			\end{figure}
111	dfer	1.1
112	dfer	1.12	\begin{figure} [tpb]
113			\begin{center}
114			\includegraphics[width=\textwidth,height=.3\textheight]{part3/case_studies/global_oce_estimation/Tutorial_fig.eps}
115			\caption{Initial annual mean surface heat flux (top right in W.m$^{-2}$) and
116			adjustment obtained at iteration 15 (bottom right). Averaged difference
117			between model and observed potential temperatures at the surface (in $^\circ$C)
118			before optimization (iteration 0, top right) and after optimization
119			(iteration 15, bottom right). Contour intervals for heat flux and temperature
120			are 25~W.m$^{-2}$ and 1$^\circ$C, respectively. A positive flux is out of the
121			ocean.}
122			\label{Results}
123			\end{center}
124			\end{figure}
125	dfer	1.1
126	dfer	1.10	\subsection{Implementation of the control variable and the cost function}
127	dfer	1.1
128	dfer	1.12	One of the goal of this tutorial is to illustrate how to implement a new
129			control variable. Most of this is fairly generic and is done in the ctrl
130			and cost packages found in the pkg/ directory. The modifications can be
131			tracked by the CPP option ALLOW\_HFLUXM\_CONTROL or the comment
132			cHFLUXM\_CONTROL. The more specific modifications required for the experiment
133			are found in verification/tutorial\_global\_oce\_optim/code\_ad. Here follows
134			a brief description of the implementation.
135	dfer	1.1
136	dfer	1.3	\subsubsection{The control variable}
137
138	dfer	1.10	The adjustment $Q_\mathrm{netm}$ is activated by setting
139			ALLOW\_HFLUXM\_CONTROL to "define" in ECCO\_OPTIONS.h.
140	dfer	1.1
141	dfer	1.10	It is first implemented as a ``normal'' forcing variable. It is defined in
142			FFIELDS.h, initialized to zero in ini\_forcing.F, and then used in
143			external\_forcing\_surf.F. $Q_\mathrm{netm}$ is made a control variable in
144			the ctrl package by modifying the following subroutines:
145	dfer	1.1
146	dfer	1.3	\begin{itemize}
147	dfer	1.10	\item ctrl\_init.F where $Q_\mathrm{netm}$ is defined as the control variable
148			number 24,
149	dfer	1.1
150	dfer	1.10	\item ctrl\_pack.F which writes, at the end of each iteration, the sensitivity
151			of the cost function $\partial J/\partial Q_\mathrm{netm}$ in to a file to be
152			used by the line-search algorithm,
153	dfer	1.1
154	dfer	1.10	\item ctrl\_unpack.F which reads, at the start of each iteration, the updated
155	dfer	1.12	adjustment as provided by the line-search algorithm,
156	dfer	1.1
157	dfer	1.12	\item ctrl\_map\_forcing.F in which the updated adjustment is added to the
158	dfer	1.10	first guess $Q_\mathrm{netm}$.
159	dfer	1.3	\end{itemize}
160	dfer	1.1
161	dfer	1.10	Note also some minor changes in ctrl.h, ctrl\_readparams.F, and ctrl\_dummy.h
162			(xx\_hfluxm\_file, fname\_hfluxm, xx\_hfluxm\_dummy).
163	dfer	1.1
164	dfer	1.3	\subsubsection{Cost functions}
165
166	dfer	1.10	The cost functions are implemented using the {\it cost} package.
167	dfer	1.1
168	dfer	1.3	\begin{itemize}
169	dfer	1.1
170	dfer	1.10	\item The temperature cost function $J_1$ which measures the drift of the mean
171	dfer	1.12	model temperature from the Levitus climatology is implemented in cost\_temp.F.
172	dfer	1.10	It is activated by ALLOW\_COST\_TEMP in ECCO\_OPTIONS.h. It requires the mean
173			temperature of the model which is obtained by accumulating the temperature in
174			cost\_tile.F (called at each time step).
175			The value of the cost function is stored in {\it objf\_temp} and its weight
176			$\lambda_1$ in {\it mult\_temp}.
177
178			\item The heat flux cost function, penalizing the departure of the surface
179			heat flux from observations is implemented in cost\_hflux.F, and activated by
180	dfer	1.12	the key ALLOW\_COST\_HFLUXM in ECCO\_OPTIONS.h. The value of the cost
181	dfer	1.10	function is stored in {\it objf\_hfluxm} and its weight $\lambda_2$ in
182			{\it mult\_hfluxm}.
183	dfer	1.1
184	dfer	1.3	\item The subroutine cost\_final.F calls the cost\_functions subroutines
185	dfer	1.10	and make the (weighted) sum of the various contributions.
186	dfer	1.1
187	dfer	1.10	\item The various weights used in the cost functions are read in
188			cost\_weights.F. The weight of the cost functions are read in
189			cost\_readparams.F from the input file data.cost.
190	dfer	1.1
191	dfer	1.3	\end{itemize}
192
193	dfer	1.5
194			\subsection{Code Configuration}
195			\label{www:tutorials}
196			\label{SEC:eg_fourl_code_config}
197
198	dfer	1.10	The model configuration for this experiment resides under the directory
199			{\it verification/tutorial\_global\_oce\_optim/}. The experiment files in
200			code\_ad/ and input\_ad/ contain the code customizations and parameter
201			settings. Most of them are identical to those used in the Global Ocean
202	jmc	1.13	( experiment {\it tutorial\_global\_oce\_latlon}). Below, we describe some of
203	dfer	1.12	the customizations required for this experiment.
204	dfer	1.10
205			\subsubsection{Compilation-time customizations in {\it code\_ad/}}
206
207	dfer	1.12	In ECCO\_CPPOPTIONS.h:
208	dfer	1.10
209			\begin{itemize}
210	dfer	1.12	\item define ALLOW\_ECCO\_OPTIMIZATION
211	dfer	1.10
212	dfer	1.12	\item define ALLOW\_COST, ALLOW\_COST\_TEMP, and ALLOW\_COST\_HFLUXM
213	dfer	1.10
214	dfer	1.12	\item define ALLOW\_HFLUXM\_CONTROL
215	dfer	1.10	\end{itemize}
216
217			\subsubsection{Running-time customizations in {\it input\_ad/}}
218
219			\begin{itemize}
220
221	dfer	1.12	\item {\it data}: note the smaller {\it cg2dTargetResidual} than in the
222			forward-only experiment,
223	dfer	1.10
224	dfer	1.12	\item {\it data.optim} specifies the iteration number,
225	dfer	1.10
226			\item {\it data.ctrl} is used, in particular, to specify the
227			name of the sensitivity and adjustment files associated to a control
228	dfer	1.12	variable,
229	dfer	1.10
230	dfer	1.12	\item {\it data.cost}: parameters of the cost functions, in particular
231			{\it lastinterval} specifies the length of time-averaging for the model
232			temperature to be used in the cost function (\ref{cost_temp}),
233	dfer	1.10
234	dfer	1.12	\item {\it data.pkg}: note that the Gradient Check package is turned on by
235			default (useGrdchk=.TRUE.),
236	dfer	1.10
237			\item {\it Err\_hflux.bin} and {\it Err\_levitus\_15layer.bin} are the
238			files containing the heat flux and potential temperature uncertainties,
239			respectively.
240	dfer	1.5
241	dfer	1.10	\end{itemize}
242	dfer	1.5
243	dfer	1.3	\subsection{Compiling}
244	dfer	1.1
245			The optimization experiment requires two executables: 1) the
246	dfer	1.10	MITgcm and its adjoint ({\it mitgcmuv\_ad}) and 2) the line-search
247			algorithm ({\it optim.x}).
248	dfer	1.1
249	edhill	1.6	\subsubsection{Compilation of MITgcm and its adjoint: {\it mitcgmuv\_ad}}
250	dfer	1.1
251			Before compiling, first note that, in the directory code\_ad/, two files
252			must be updated:
253	dfer	1.5	\begin{itemize}
254			\item code\_ad\_diff.list which lists new subroutines to be compiled
255	dfer	1.10	by the TAF software (cost\_temp.F and cost\_hflux.F here),
256	dfer	1.1
257	dfer	1.5	\item the adjoint\_hfluxm files which provides a list of the control variables
258	dfer	1.10	and the name of cost function to the TAF software.
259	dfer	1.5
260			\end{itemize}
261	dfer	1.1
262			Then, in the directory build\_ad/, type:
263	dfer	1.5	\begin{verbatim}
264			% ../../../tools/genmake2 -mods=../code\_ad -adof=../code\_ad/adjoint\_hfluxm
265			% make depend
266			% make adall
267			\end{verbatim}
268	dfer	1.10	to generate the MITgcm executable {\it mitgcmuv\_ad}.
269	dfer	1.1
270	edhill	1.6	\subsubsection{Compilation of the line-search algorithm: {\it optim.x}}
271	dfer	1.1
272	dfer	1.10	This is done from the directories lsopt/ and optim/ (under MITgcm/). In lsopt/,
273			unzip the blash1 library adapted to your platform, and change the Makefile
274			accordingly. Compile with:
275	dfer	1.5	\begin{verbatim}
276			% make all
277			\end{verbatim}
278			(more details in lsopt\_doc.txt)
279	dfer	1.1
280	dfer	1.10	In optim/, the path of the directory where {\it mitgcm\_ad} was compiled
281			must be specified in the Makefile in the variable INCLUDEDIRS. The file name
282			of the control variable (xx\_hfluxm\_file here) must be added to the name list
283			read by optim\_num.F. Then use
284	dfer	1.5	\begin{verbatim}
285			% make depend
286			\end{verbatim}
287			and
288			\begin{verbatim}
289			% make
290			\end{verbatim}
291	edhill	1.6	to generate the line-search executable {\it optim.x}.
292	dfer	1.1
293	dfer	1.3	\subsection{Running the estimation}
294	dfer	1.1
295	dfer	1.10	Copy the {\it mitgcmuv\_ad} executable to input\_ad/ and {\it optim.x} to the
296			subdirectory input\_ad/OPTIM/. Move into input\_ad/. The first iteration is
297			somewhat particular and is best done "by hand" while the following iterations
298			can be run automatically (see below). Check that the iteration number is set
299			to 0 in data.optim and run the MITgcm:
300	dfer	1.3	\begin{verbatim}
301			% ./mitgcmuv_ad
302			\end{verbatim}
303	dfer	1.1
304	dfer	1.5	The output files adxx\_hfluxm.0000000000.* and xx\_hfluxm.0000000000.* contain
305	dfer	1.12	the sensitivity of the cost function to $Q_\mathrm{netm}$ and the adjustment
306	dfer	1.10	to $Q_\mathrm{netm}$ (zero at the first iteration), respectively. Two other
307	dfer	1.12	files called costhflux\_tut\_MITgcm.opt0000 and ctrlhflux\_tut\_MITgcm.opt0000
308			are also generated. They essentially contain the same information as the
309			adxx\_.hfluxm* and xx\_hfluxm* files, but in a compressed format. These two files
310			are the only ones involved in the communication between the adjoint model
311			{\it mitgcmuv\_ad} and the line-search algorithm {\it optim.x}. Only at the first
312			iteration, are they both generated by {\it mitgcmuv\_ad}. Subsenquently,
313			costhflux\_tut\_MITgcm.opt$n$ is an output of the adjoint model at
314	dfer	1.10	iteration $n$ and an input of the line-search. The latter returns an updated
315	dfer	1.12	adjustment in ctrlhflux\_tut\_MITgcm.opt$n+1$ to be used as an input of the
316			adjoint model at iteration n+1.
317	dfer	1.10
318	dfer	1.12	At the first iteration, move costhflux\_tut\_MITgcm.opt0000 and
319			ctrlhflux\_tut\_MITgcm.opt0000 to OPTIM/, move into this directory and link data.optim
320			and data.ctrl locally:
321			\begin{verbatim}
322			% cd OPTIM/
323			% ln -s ../data.optim .
324			% ln -s ../data.ctrl .
325			\end{verbatim}
326			The target cost function {\it fmin} needs to be specified too: as a rule of thumb,
327			it should be about 0.95-0.90 times the value of the cost function at
328	dfer	1.10	the first iteration. This value is only used at the first iteration and does
329			not need to be updated afterward. However, it implicitly specifies the
330			``pace'' at which the cost function is going down (if you are lucky and it does
331			indeed diminish!). More in the ECCO section maybe ?
332	dfer	1.4
333			Once this is done, run the line-search algorithm:
334			\begin{verbatim}
335			% ./optim.x
336			\end{verbatim}
337	dfer	1.12	which computes the updated adjustment for iteration 1, ctrlhflux\_tut\_MITgcm.opt0001.
338	dfer	1.4
339	edhill	1.6	The following iterations can be executed automatically using the shell
340	dfer	1.10	script {\it cycsh} found in input\_ad/. This script will take care of changing
341			the iteration numbers in the data.optim, launch the adjoint model, clean and
342	dfer	1.12	store the outputs, move the costhflux* and ctrlhflux* files, and run the
343	dfer	1.10	line-search algorithm. Edit {\it cycsh} to specify the prefix of the
344			directories used to store the outputs and the maximum number of iteration.
345	dfer	1.12